Some universities require that each graduate student have a minor. I wonder whether it is helpful in the long run? For example, if one has PhD in mathematics with a minor in philosophy, would it be helpful to obtain a position in logic in a department of philosophy? [Probably related to this question.]
As you kind of alluded to in your example, the answer to this question depends a lot on what kind of "long run" you are planning to pursue.
First, in a career where your minor subject is irrelevant, then no, your minor likely will not matter. Sounds obvious, but this is likely the case for many people.
Further, in a career where your minor subject becomes your main focus, such as a mathematician transitioning to a position in philosophy, then you will likely need to demonstrate your qualifications with more than just your PhD minor. Your minor may be noticed when looking for your first academic appointment, but in the long run, it will be much less helpful than publications in philosophy, recommendations from fellow academics in philosophy, coursework in philosophy subjects etc.
There are a few situations where the minor could be helpful, such as if you are applying for an entry-level job in some interdisciplinary field. However, there are many more situations where the minor is either irrelevant or insufficient to prove your qualifications.
Based on my own experience (i.e., n = 1), a graduate minor during my PhD helped me a lot. In my case, I earned a PhD in an environmental field and earned a minor in applied math. The minor helped me demonstrate quantitative skills. I have also seen other people who earn either a minor or master's degree during the course of their PhD in statistics and this help them a great deal.
More broadly, a graduate minor can be helpful if it compliments your primary field of study. For example, a PhD with a major in wildlife ecology with a minor in biology would likely not be beneficial, but a PhD with a major in wildlife ecology with a minor in sociology might be helpful if one wants to do human dimensions research.